1. Chapter 14: Simple harmonic motion
Physics 223, Fall2018

2. Oscillatory Motion
An oscillator is an object or system of objects that undergoes periodic oscillatory motion or behavior. Example: a mass and spring system.
Characteristics:
Oscillatory motion is about some (energy = 0) equilibrium position;
Oscillatory motion is periodic, with a definite period T or cycle time.
Frequency f has units of hertz.

3. Example
What is the oscillation period for the broadcast of a 100 MHz FM radio station?

4. An Experiment
Consider Fig. (a). An air-track glider attached to a spring. The glider is pulled a distance A from its rest position and released.
Fig. (b) shows a graph of the motion of the glider, as measured each 1/20 of a second.
The graphs on the right show the position and velocity of the glider from the same measurements. We see that A=0.17 m and T=1.60 s. Therefore the oscillation frequency of the system is f = Hz
This is an example of simple harmonic motion (SHM).

5. Describing SHM

6. SHM Example
An air-track glider is attached to a spring, pulled 20 cm to the right, and released at t = 0. It makes 15 oscillations in 10 s.
What is the period T of oscillation?
What is the object’s maximum speed?
What is the position and velocity at t = 0.80 s? /s

7. Example: Finding the time
A mass on a spring oscillating in simple harmonic motion (SHM) starts at x=A at t=0 and has period T.
At what time, as a fraction of T, does the object first pass through x = ½A?

8. SHM and Circular Motion
Uniform circular motion projected into one dimension is simple harmonic motion (SHM).
Consider a particle rotating ccw, with the angle f increasing linearly with time:

9. The Phase Constant But what if f is not zero at t =0?
A phase constant f ¹ 0 means that the rotation starts at a different point on the circle, implying different initial conditions.

10. Phases and Oscillations1 2 3
Here are three examples of differing initial conditions:
f0 = p/3, implying x0 = A/2 and moving to the left (v<0);
f0 = –p/3, implying x0 = A/2 and moving to the right (v>0);
f0 = p, implying x0 = –A and momentarily at rest (v=0).

11. Using Initial Conditions
A mass on a spring oscillates with a period of 0.80 s and an amplitude of 10. cm. At t = 0, it is 5.0 cm to the right of equilibrium and moving to the left.
What is the position and direction of motion at t = 2.0 s?
Wait! This doesn’t make sense – two seconds is a two-and-a-half periods, the mass should be moving to the right!

12. Using Initial Conditions
A mass on a spring oscillates with a period of 0.80 s and an amplitude of 10. cm. At t = 0, it is 5.0 cm to the right of equilibrium and moving to the left.
What is the position and direction of motion at t = 2.0 s?
Careful about radians!

13. Energy of SHM

14. Velocity and Amplitude of SHM

15. Example: Using Conservation of Energy
A 500 g block on a spring is pulled a distance of 20. cm and released. The subsequent oscillation is measured to have a period of 0.80 s.
At what position(s) is the block’s speed 1.0 m/s?

16. SHM model for atoms in a solid
(a) If the atoms in a molecule do not move too far from their equilibrium positions, a graph of potential energy versus separation distance between atoms is similar to the graph of potential energy versus position for a simple harmonic oscillator. (b) The forces between atoms in a solid can be modeled by imagining springs between neighboring atoms.

17. Dynamics of SHM
The acceleration is proportional to the negative of the displacement at any time.

18. The Equation of Motion in SHM
(Whenever the restoring force (or acceleration) is proportional to position, you have simple harmonic motion.)
This is the equation of motion for the system. It is a homogeneous linear 2nd order differential equation.

19. Solving the Equation of Motion
Q: How do you solve this differential equation?
A: The standard method - guess the solution and see if it works.
It works!

20. Example
At t=0, a 500 g block oscillating on a spring is observed to be moving to the right at x=15 cm. At t=0.30 s, it reaches its maximum displacement of 25 cm.
Draw a graph of the motion for one cycle.
At what time in the first cycle is x = 20 cm?

21. Vertical Oscillations
Now consider a mass m hanging from a spring with constant k. When the mass is attached, the spring stretches by DL. In this configuration, the equilibrium position is given by: m -

22. Example: A Vertical Oscillator
A 200 g block hangs from a spring with constant k = 10 N/m. The block is pulled down to a point where the spring is 30 cm longer than its unstretched length, then released.
Where is the block and what is its velocity 3.0 s later?

23. The Pendulum
at =
This is not a homogeneous linear 2nd order differential equation describing SHM because the acceleration is proportional to the sine of the angular displacement and not just the angular displacement itself.

24. The Small Angle Approximation
The pendulum equation is:
For small angles l is almost equal to r.
Taylor series expansion:
sin x = x – x3/3! + x5/5! – …
If x << 1 radian, then sin x » x.
Small angle approximation works well if theta is less than 10 degrees.
This is a homogeneous linear 2nd order differential equation describing SHM, but only for small swings (s<<L, or q less than 10 degrees or so). The angular frequency is w = (g/L)½, and the period is T = 2p(L/g)½.

25. Example: A Pendulum Clock
What length L must a pendulum have to give an oscillation period T of exactly 1 s?

26. Example: The Maximum Angle of a Pendulum
A 300 g mass on a 30 cm long string oscillates as a pendulum. It has a speed of 0.25 m/s as it passes through the lowest point.
What maximum angle qmax does the pendulum have?

27. General Physical Pendulum
Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, and that we know where the CM is located and what the moment of inertia I about the axis is.
The torque about the rotation (z) axis for small  is (sin  q )  = -Mgd -MgR
z-axis R  x CM d S  Mg
where
 = 0 cos(t + )

28. Example: Walking
The walking speed of T. rex can be estimated from the leg length L and the stride length S.

29. For a physical pendulum, so T =
Suppose the natural walking pace of T. rex is equal to the period of the leg. We can estimate this by assuming the leg is a uniform rod pivoted at the hip joint. Fossil evidence shows T Rex had a leg length L of 3.1 m and a stride length S of 4.0 m.
For a physical pendulum, so T =
The moment of inertia for a uniform rod of length L about one end is
I =
So T =
which is proportional to Note that R = L/2 since the center of mass of the rod is at the midpoint of the rod.

30. Each period (a back and forth swing of the leg) corresponds to two steps. This means the walking pace in steps per unit time is twice the oscillation frequency. Since f = 1/T, the walking pace is proportional to
For T. rex, T =
And the walking speed is v = = 2.8 m/s = 6.2 mi/h
A better estimate would take into account the fact that there is more mass in the leg between the knee and the hip than between the knee and the foot. This puts the center of mass of the leg at say L/4 not L/2. This would make I much smaller, say something like I =
and the walking speed would be considerably faster.

31. Physical Pendulum Example
A pendulum is made by hanging a thin hoola-hoop of diameter D on a small nail. What is the angular frequency of oscillation of the hoop for small displacements? (ICM = mR2 for a hoop)
pivot (nail)
(a)
(b)
(c) D

32. Solution
The angular frequency of oscillation of the hoop for small displacements will be given by
Use parallel axis theorem: I = Icm + mR2
pivot (nail)
= mR2 + mR2 = 2mR2
cm x R So m

33. Conditions for SHM
For the horizontal mass+spring system, Fsp = -kx. This linear dependence of the restoring (-) force on position x is the key requirement for simple harmonic motion.
In a sense, once we have solved for the simple harmonic motion behavior of the horizontal mass+spring system, we have solved all problems involving simple harmonic motion.
Moreover, even when the restoring force is nonlinear (e.g., the pendulum), for sufficiently small displacements it can be treated as a linear system (small angle approximation).
Therefore, we often treat atoms, molecules, and nuclei as small harmonic oscillators, even when the restoring forces involved may be very different from Hooke’s Law forces.

34. Identifying and Analyzing SHM
If the net force acting on a particle is a linear restoring force, the motion will be simple harmonic motion (SHM) around the equilibrium position.
The position x as a function of time is x(t) = A cos(wt + f0). The velocity v as a function of time is v(t) = -wA sin(wt + f0). These equations are written here in terms of x, but they can be written in terms of any other appropriate variables (y, q, etc.)
The amplitude A and the phase constant f0 are determined by the initial conditions, through x0 = A cos f0 and v0x = -wA sin f0.
The angular frequency w and the period T depend on the physics of the particular situation but are independent of A and f0.
Mechanical energy is conserved. Thus, ½mvx2 + ½kx2 = ½kA2 = ½mvmax2. Energy conservation provides a relationship between position and velocity that is independent of time.